The product, the result of multiplying numbers with repeating 1s, has this beautiful symmetry. The products are palindromes, which means that they are the same whether written forwards or backwards.

2 marks

1.1. What happens if we add one more 1?

What is 1111111111 × 1111111111?

Of course, you can type this into a calculator, but first try to follow the pattern and work out what the answer might be… then check it with a calculator.

Correct Solution: 1234567900987654321

If you follow the pattern, the result should be:

12345678910987654321.

The 10 fits in the middle, but we can’t write 10 where there is only space for 1 digit. Instead, we could put a 0 where the 10 is, and carry the 1 over to the column on the left, which gives us:

123456789+10987654321.

But we can’t write (9 + 1), which is 10, where there is only space for 1 digit. Instead, we could put a 0 in that column and carry the 1 over to the column on the left, which gives us:

1234567900987654321.

2. Building brilliant paper planes

Here is an interview with John Collins, who broke the world distance record for paper planes. Although John designs and builds the planes, he does not throw them.

Listen carefully, as I will be asking a question about the thrower.

1 mark

2.1 The thrower is an ex-sportsman, but which sport did he play?

American football

Baseball

Cricket

Darts

Elephant Polo (yes, it’s a real sport, and more importantly it means that the answers start with an A, B, C, D and E).

The answer is American football, because the paper plane engineer keeps talking about the thrower being a ‘quarterback’.

If you didn’t know that a quarterback is an American footballer, then a quick google would have solved the problem.

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1 A 5 cm × 5 cm square is cut into five pieces, as shown.

Each cut is a sequence of identical copies of the same shape but pointing up, down, left or right.

Which piece has the longest perimeter?

A

B

C

D

E

Show Hint (–1 mark)

–1 mark

If we look at pieces B and C, they have roughly the same perimeter – let’s call it 12 lengths. However, B has a longer perimeter than C, because more of B’s sides are crooked, and crooked sides are longer than straight sides.

We let the length of each crooked line be x cm. Since this crooked line is longer than the side length of one of the squares, 1 < x.